Student Feature - Stephanie Dodson

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I am interested in research questions in dynamical systems motivated by applications in biology. Currently, my work involves the formation and stability of spatiotemporal patterns and how they can be used to understand hidden relationships in the natural world. I am a fifth-year PhD candidate in the Division of Applied Mathematics at Brown University, where I work with Dr. Björn Sandstede to understand the stability of spiral waves.

As an undergraduate in mathematics and physics at the University of Massachusetts, Amherst, my undergraduate coursework introduced me to a variety of applications in physics and biology that I found fascinating. I participated in a bioinformatics Research Experience for Undergraduates, computational biology summer internship, and conducted honors thesis research in particle physics. Each experience has shaped my interests and led me to where I am today.

Spiral wave patterns are frequently observed in nature, including in cardiac arrhythmias and chemical oscillations. The irregular heart rhythms associated with ventricular tachycardia have been linked to spiral waves in cardiac electrical activity. When these spirals destabilize, chaotic electrical activity can take over, leading to deadly ventricular fibrillation. As a precursor to fibrillation, spirals have been observed to undergo alternans – an instability that causes a beat to beat variation in the action potential duration. Another instability, which takes the form of a stationary line defect, is seen in chemical reactions.

Both alternans and line defects can be thought of as period-doubling instabilities in that the bifurcated spirals need to be rotated twice to complete a period. Despite the similarities in temporal behavior, the mechanisms driving the instabilities are quite different – alternans appear to be driven by the spiral core whereas line defects emerge from the outer boundary. The properties of alternans and line defects are qualitatively captured in the Karma and Rössler models, respectively. For my PhD thesis, I am working with Dr. Sandstede to investigate the instabilities in these reaction-diffusion systems using techniques from dynamical systems and spectral theory. Additionally, we want to understand how the behaviors and spectra differ on infinite versus bounded domains.

Through the 2018 NSF Graduate Research Internship Program, I began working with Dr. Elliott Hazen and Dr. Steven Bograd at the NOAA Environmental Research Division to study the spatiotemporal distribution of blue whales off the California coast. Blue whales are a highly dynamical species, and we developed an agent-based model to investigate how short term decisions along with environmental and prey conditions impact the migratory behavior. The model accurately captures the spring-summer northward migration and yearly differences in the spatiotemporal distribution driven by variations in prey abundance. We are now exploring drivers of the fall southward migration.

Outside of research, I am passionate about sharing my enthusiasm for mathematics through mentoring and teaching. I was the President of the Brown University Association for Women in Mathematics Student Chapter, and currently an active organizer for the group. I plan panels, math talks, and other professional development events centered around supporting underrepresented groups and highlighting mathematical careers and research applications. Additionally, I participate in workshops and courses to learn about inclusive, student-centered teaching practices. I implemented active-based learning practices during summer 2017 when I had the opportunity to teach an introductory differential equations course at Brown University. In the course, I also included problems from engineering, physics, biology, immunology, and economics to showcase the variety of applications and engage the students.

I enjoy viewing, learning about, and connecting diverse application areas with the lens of mathematics. In future research directions, I want to continue researching patterns found in nature and the mechanisms that form them, including exploring questions related to ecology, conservation, and climate change. Furthermore, I look forward to sharing my excitement for mathematics with others at conferences and through teaching, mentoring, and outreach programs.